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Math-2863 · mathematics · Probability

The joint GFF and cluster limit of critical Ashkin–Teller currents

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each fixed $J>0$ and $U\leq J$ on $sinh(2J)=\exp (-2U)$, do the Ashkin–Teller current height, wired and free cluster collections, and distinguished boundary component converge jointly on every admissible Jordan-domain lattice approximation to the source-normalized GFF, its associated continuum cluster collections, and its specified symmetric two-valued set? Use $H^s_loc$ for every $s<-1$, cluster matching, and Hausdorff convergence for the distinguished component.

Short English statement

For each fixed $J>0$ and $U\leq J$ on $sinh(2J)=\exp (-2U)$, do the Ashkin–Teller current height, wired and free cluster collections, and distinguished boundary component converge jointly on every admissible Jordan-domain lattice approximation to the source-normalized GFF, its associated continuum cluster collections, and its specified symmetric two-valued set? Use $H^s_loc$ for every $s<-1$, cluster matching, and Hausdorff convergence for the distinguished component.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2863-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2863

Along the critical Ashkin--Teller line, the height and both nested current-cluster systems are claimed to converge jointly to one Gaussian field and its canonical recursive two-valued local sets. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

The joint GFF and cluster limit of critical Ashkin–Teller currents

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Along the critical Ashkin--Teller line, the height and both nested current-cluster systems are claimed to converge jointly to one Gaussian field and its canonical recursive two-valued local sets.

Mathematical statement of the claimed result

Result 1.
Fix $J>0,U\le J$ on $\sinh(2J)=e^{-2U}$, and set $t=(1+e^{4U})^{-1/2}$, $c=1/t$, $g=8\arcsin(c/2)/\pi$, $a=\pi\sqrt g/2$. In a bounded Jordan domain with uniformly approximating simple lattice polygons, use the specified wired-primal/free-dual Ashkin--Teller current coupling: spin pairs have no simultaneous primal/dual disagreement and weight $t^{N(\tau)+N(\tau^\dagger)}$; when neither disagrees, edge states $(1,0),(0,1),(0,0)$ have probabilities $1-t,1-t,2t-1$. The height is pinned zero on the boundary and has corner increments $\tfrac12\tau(v)\tau^\dagger(f)$. The claimed joint limit is \[ (H_\delta,\mathcal C_\delta^{\mathrm w},\mathcal C_\delta^{\mathrm f},C_{\partial,\delta}) \Longrightarrow(h/(\pi\sqrt g),\mathcal C^{\mathrm w},\mathcal C^{\mathrm f},A_{-a,a}(h)). \] Here $h$ is the zero-boundary GFF with logarithmic singularity $-\log|z-w|$, and $A_{-a,a}$ is its canonical two-valued local set. The continuum wired/free cluster families are the recursive local-set constructions from this same $h$. Heights converge locally in $H^s$, $s<-1$; cluster matching and the distinguished boundary component use Hausdorff geometry, retaining every positive-diameter cluster. Conditional on spins, different edge-pair samples are independent. Dual disagreement forces the edge pair $(1,0)$, and primal disagreement forces $(0,1)$. Primal spins are fixed to $+1$ on the boundary, the boundary-cycle primal edges are open, and dual boundary spins are free. The height distribution is smeared atomically: \[ H_\delta(\varphi)=\delta^2\sum_{v\in V_\delta}H_\delta(v)\varphi(v). \]

Outline of the argument

Exact cuts and spatial comparisons establish reflection-positive covariance and Gaussian Laplace screening. Stopped local-set exploration keeps the current clusters coupled to the same field, and iteration retains every nesting level together with the wired boundary cluster, including the four-state endpoint.

Formal verification

Family 233 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

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